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Some basic concept Vibration of Structures are Analysis of Membrane Over Enclosure, Applications of Modal Solution, Applications of Wave Solution, Approximate Analysis of Membranes, Approximate Methods, Axial and Torsional Vibrations, Beam Models, Circular Membrane.Main points of this lecture are: Approximate Methods, Rayleigh Method, Rayleigh-Ritz Method, Ritz Method, Approximate Modal Analysis, Non-Uniform Bar, Admissible Functions, Young's Modulus, Rayleigh Quotient, Minimization Problem, Ele
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Contents:
Keywords: Approximate modal analysis, Non-uniform bar, Rayleigh quo- tient, Ritz expansion, Admissible functions
Exact solutions of the modal analysis problem are usually too cumbersome to obtain. In such situations, approximate methods can provide sufficiently accurate results to serve the purpose.
Rayleigh’s method can be used to estimate the lowest (or fundamental) frequency of a self-adjoint (conservative) continuous system. Consider a bar of length l, density ρ, having an area of cross-section A and Young’s modulus E undergoing axial vibrations. The total mechanical energy of the system comprising the kinetic and potential energies is given by E = T + V =^12
∫ (^) l 0 ρAu
(^2) ,t(x, t) dx +^1 2
∫ (^) l 0 EAu
(^2) ,x(x, t) dx. (1)
Assuming that the system is vibrating in one of its eigenmodes, we can write the solution as u(x, t) = U(x) cos ωt, where U(x) is an eigenfunction of the system, and ω the corresponding circular natural frequency. Substituting this
2
A 0 A 0 / 4
A(x) = A 0 (1 − x/ 2 l)^2 l
x
u(x, t) ρ, E
Figure 1: A tapered circular bar
mined later. Computation of the numerator of the Rayleigh quotient in (3) yields ∫ (^) l 0 EAF^
′ (^2) (x) dx = EA^0 l α
24 (2α^ −^ 1)2α^ + 4(2α^ + 1) (2α − 1)2α(2α + 1) ,^ (5)
where it is required that α > 1 /2 for the definite integral in the above to exist and be positive definite. Further, if α < 1, then as x → 0, F ′(x) → ∞. The denominator of the Rayleigh quotient yields ∫ (^) l 0 ρAF^
(^2) (x) dx = ρA 0 l (2α^ + 1)(2α^ + 2) + 4(2α^ + 3) 4(2α + 1)(2α + 2)(2α + 3).^ (6)
Therefore, the Rayleigh quotient is obtained as
R[F (x), α] = E ρlα(2α (^2) + 3α + 2)(2α (^2) + 5α + 3) (2α^2 + 7α + 7)(2α − 1).^ (7)
We can now minimize R[F (x), α] with respect to α, which yields α ≈ 0 .93, and the fundamental circular frequency as ω 1 ≈ 2. 08303 c/l. The exact solu- tion is given by ωexact 1 = 2. 029 c/l. It may be noted that the obtained shape function F (x) after minimization cannot be used to determine the strain at the fixed end since F ′(0) = ∞. However, it gives the frequency estimate
4
within 3% of the exact value.
In Rayleigh-Ritz method, we consider the expansion of a mode shape in terms of N linearly independent admissible functions Ui(x), i = 1, 2 ,... , N in the form U(x) =
i=
αiUi(x), (8)
where αi are unknown constants which are to be chosen suitably to minimize the Rayleigh quotient. Substituting (8) in the Rayleigh quotient (3) leads to
ω^2 =
∑Ni,j=1^ αiαj^ kij i,j=1 αiαj^ mij
= α T (^) Kα αT^ Mα,^ (9) where kij =
∫ (^) l 0 EAU i′ U j′ dx,^ and^ mij^ =^ ∫^ l 0 ρAUiUj^ dx,^ (10) α is the vector of the αi, and T in the superscript indicates transposition. Now the minimization condition of the Rayleigh quotient can be written as ∂ ∂αp
( (^) αT (^) Kα αT^ Mα
= 0, p = 1, 2 ,... , N, ⇒ αT^ Mα
(∂αT (^) Kα ∂α
− αT^ Kα
(∂αT (^) Mα ∂α
⇒ Kα −
( (^) αT (^) Kα αT^ Mα
Mα = 0 , ⇒ (K − ω^2 M)α = 0 (using (9)). (11)
Thus, the extremization condition for Rayleigh’s quotient (formed using a fi- nite expansion) for a continuous system leads to the eigenvalue problem of a 5
where
M =
∫ (^) l 0 ρAHH
T (^) dx, and K =^ ∫^ l 0 EAH
′H′T (^) dx. (15)
The discrete equations of motion are obtained as
Mp¨ + Kp = 0. (16)
It may be observed from (15) that both M and K are symmetric. They are also positive definite since ρA and EA are positive functions, and the admissible functions chosen in the expansion (13) are linearly independent. Consider the example of axial vibrations of the tapered bar shown in Fig. 1. We can choose the admissible functions as
Hj (x) = x l
1 − 2 xl
)j− 1 , j = 1, 2 ,... , N, (17)
since the geometric boundary conditions, Hj (0) = 0, are exactly satisfied. However, as can be checked, the natural boundary conditions, H (^) j′ (l) = 0, are not satisfied. Considering only two admissible functions in the expansion (13), and following the steps discussed above, (16) takes the form
ρA 0 l
{ (^) p¨ 1 p ¨ 2
{ (^) p 1 p 2
Assuming a modal solution p(t) = keiωt, (18) yields the eigenvalue problem
[−ω^2 M + K]k = 0, (19) 7
from which the characteristic equation is obtained as 81 7 ω
(^4) − 394 c^2 l^2 ω
(^2) + 1455c^2 l^2 = 0.^ (20) The first two approximate circular natural frequencies of longitudinal vibra- tion are obtained from (20) as ω 1 R = 2. 053 c/l and ωR 2 = 5. 462 c/l. The exact circular eigenfrequencies were obtained previously as ω 1 exact = 2. 029 c/l, and ω 2 exact = 4. 913 c/l. The eigenvectors are obtained from (19) as
k 1 =
, and k 2 =
Using these eigenvectors, along with (17), in (13), we get the approximate eigenfunctions as
U 1 (x) = HT^ k 1 = x l + 1. 457 x l
1 − 2 xl
and U 2 (x) = HT^ k 2 = x l − 1. 505 x l
1 − 2 xl
It may be observed from the comparison in Fig. 2 that the first mode shape is determined reasonably accurately. However, the second mode shape is in considerable error. For a better approximation of the second mode shape, one must take more terms in the expansion (13). In general, the error in deter- mination of the eigenfrequencies is less than that in the determination of the eigenfunctions. Further, the eigenfrequencies are always overestimated. This is expected since we are approximating an infinite degrees of freedom system by a finite degrees of freedom system, there by increasing the stiffness of the 8
